Försättsblad till skriftlig tentamen vid Linköpings universitet

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1 Försättsblad till skriftlig tentamen vid Linköpings universitet Datum för tentamen Sal (1) KÅRA T1 & T (Om tentan går i flera salar ska du bifoga ett försättsblad till varje sal och ringa in vilken sal som avses) Tid 8: 13: Kurskod TSRT3 Provkod TEN1 Kursnamn/benämning Reglerteknik Institution ISY Antal uppgifter som ingår 5 i tentamen Jour/kursansvarig Tianshi Chen (Ange vem som besöker salen) Telefon under skrivtiden , Besöker salen cirka kl. 9:, 11:, 1: Kursadministratör/ kontaktperson Carina Lindström, , Carina.E.Lindstrom@liu.se (Namn, telefonnummer, mejladress) Tillåtna hjälpmedel Övrigt Vilken typ av papper Rutigt ska användas, rutigt eller linjerat Antal exemplar i påsen 1. T. Glad & L. Ljung: Reglerteknik. Grundläggande teori eller liknande bok i reglerteknik. Tabeller och formelsamlingar, t.ex.: L. Råde & B. Westergren: Mathematics handbook, C. Nordling & J. Österman: Physics handbook, S. Söderkvist: Formler & tabeller 3. Miniräknare utan färdiga program Inläsningsanteckningar får finnas i böckerna.

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3 TENTAMEN I TSRT3 REGLERTEKNIK SAL: KÅRA T1 & T TID: kl. 8: 13: KURS: TSRT3 Reglerteknik PROVKOD: TEN1 INSTITUTION: ISY ANTAL UPPGIFTER: 5 ANSVARIG LÄRARE: Tianshi Chen, tel , BESÖKER SALEN: cirka kl. 9:, 11:, 1: KURSADMINISTRATÖR: Carina Lindström, , Carina.E.Lindstrom@liu.se TILLÅTNA HJÄLPMEDEL: 1. T. Glad & L. Ljung: Reglerteknik. Grundläggande teori eller liknande bok i reglerteknik. Tabeller och formelsamlingar, t.ex.: L. Råde & B. Westergren: Mathematics handbook, C. Nordling & J. Österman: Physics handbook, S. Söderkvist: Formler & tabeller 3. Miniräknare utan färdiga program Inläsningsanteckningar får finnas i böckerna. LÖSNINGSFÖRSLAG: Finns på kursens websida efter skrivningens slut. VISNING av tentan äger rum , kl i Library (Room: A-598), B-huset, ingång 5, A-korridoren till höger. PRELIMINÄRA BETYGSGRÄNSER: betyg 3 3 poäng betyg 4 33 poäng betyg 5 43 poäng OBS! Lösningar till samtliga uppgifter ska presenteras så att alla steg (utom triviala beräkningar) kan följas. Bristande motiveringar ger poängavdrag. Lycka till!

4 1. (a) People with diabetes need to adjust their blood glucose level by supplying insulin, traditionally using insulin shots. Describe this from a control perspective (using the terms r(t), y(t), u(t) and model), and propose a device based on automatic control that would make shots unnecessary (no math required to solve this exercise). (3p) (b) A linear system that we are interested to control is described by the following linear ordinary differential equation (ODE) d dt y(t) + 3 d y(t) + y(t) = u(t) dt Assume y() = and d dt y(t) t= =. Then what is the transfer function of this system, i.e., Is the system stable? G(s) = Y (s) U(s) =? (p) (c) There is another linear system whose transfer function is described by G(s) = Y (s) U(s) = 1 s + s + 1 Now we insert a sinusoidal input signal to the system and the input u(t) = sin(t). What will the output y(t) be after a sufficiently long time (i.e. after transients have disappeared)? (p) (d) Consider a typical feedback control system whose block diagram is shown in Figure 1. Figur 1: Block diagram of a typical feedback control system Here, G(s) is the transfer function of the system that we are interested to control and is assumed to have no poles at the origin, and the controller F (s) is assumed to be a proportional controller, i.e., F (s) = K p with K p >. What s more, unit step signal (i.e., its amplitude is 1) is used as the reference signal r(t) inserted to the feedback control system. For K p = 1, the steady state output of the closed-loop system is 3 4. What is the steady state output of the closed-loop system if K p = 3 instead? (3p)

5 . (a) Consider the feedback control system depicted in Figure 1 again. Now assume the controller F (s) takes the form of F (s) = KF (s). The root locus with respect to K for the closed-loop system is shown in Figure. Assume that there is no cancelation of poles and zeros in the product F (s)g(s). 1 Root Locus 8 6 K=85 K = 5 4 Imaginary Axis K=3.5 K=15 K= Real Axis Figur : Given root-locus with respect to K where x denotes the start point, and o denotes the end point. (a.1) For which K-values is the system stable, and for which K- values are all poles real? (p) (a.) What will the stationary control error be if the reference signal is a step signal? (1p) (b) Now consider the feedback control of an unstable exoterm chemical process. The input is the supplied power to heat or cool the reaction. The transfer function of the chemical process G(s) is given by G(s) = s 1 (b.1) The chemical process is controlled using a P-controller and the corresponding block-diagram is shown in Figure 3. Which requirement must the gain K p in the P-controller satisfy to guarantee the stability of the closed-loop system? (p) (b.) The chemical process is not heated/cooled directly, but from 3

6 Figur 3: Block diagram of the controlled chemical process heating/cooling tank, which has transfer function G v (s) = 1 s + 4 The full transfer function of the chemical process and the heating/cooling tank is given by G(s)G v (s) = (s 1)(s + 4) Now the system we are interested to control becomes the chemical process and the heating/cooling tank. If we still use a P-controller, the corresponding block-diagram is shown in Figure 4. Draw the root-locus w.r.t to the gain K p in the P- Figur 4: Block diagram of the controlled chemical process and heating/cooling tank controller. Mark, compute and comment points of interest. (5p) 4

7 3. The uptake of a substance in the body (e.g., a medicine) can be described by the following model dq(t) = k 1 q(t) + u(t) dt dy(t) = k 1 q(t) k y(t) dt where the input u(t) is the supply of the substance, the output y(t) is the amount of the substance in the blood, and q(t) is the amount of the substance in the intestines. The constants k 1 and k describe metabolism and satisfy k 1 > k >. Here, we assume k 1 =.5 and k =.. Define a state vector x(t) for the above system as follows: x(t) = Then consider the following questions: [ ] q(t) y(t) (a) Derive the state-space model with x(t) as the state vector, i.e., determine the matrices A, B, C, D in ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (1) (1p) (b) Consider the state-space model (1). If we consider to design a state-feedback control, can we place the poles of the closed-loop state feedback control system arbitrarily? (1p) (c) Consider the state-space model (1). Design a state-feedback controller such that the closed-loop state feedback control system has poles at.1 and.1, and moreover, a step reference signal can be tracked exactly without steady state error. (5p) (d) Consider the state-space model (1). If q(t) is not accessible, design another controller to accomplish the same task in (b). (3p) 5

8 4. (a) For a given system G(s), we have designed a controller F (s) in certain ways and the block-diagram of the feedback control system is shown in Figure 5, Figur 5: Block diagram of the designed feedback control system and Figure 6 shows the Bode plot of the corresponding closedloop system G c (s) = Y (s) R(s) However, we are later told that in reality the true system is given by G (s) = G(s)(1 + α) where α >. For which α > can stability for the true closed loop system be guaranteed when the controller F (s) is used? (G and G have the same number of poles in the right half plane and both F (iω)g(iω) and F (iω)g (iω) tend to zero when ω.) (p) Bode Diagram Figur 6: Bode plot of the closed-loop system G c (s) in Problem 4 (a). 6

9 (b) In Figure 8, unit step responses for six different feedback control systems are given. In Figure 9, the Bode plots of the six feedback control systems (corresponding to closed-loop system G c (s) = Y (s)/r(s)) are given. Assume that the closed-loop system takes the following form Figur 7: The closed-loop system G c (s) and the open-loop system G o (s). Then in Figure 1, Bode plots for the six open-loop systems G o (s) are given. Pair the correct step response in Figure 8 with the correct closedloop Bode plot in Figure 9 and correct open-loop Bode plot in Figure 1. (8p) 7

10 Step Response Step Response Amplitude 1 Amplitude Time (sec) Time (sec) (a) Step response A (b) Step response B Step Response Step Response Amplitude 1 Amplitude Time (sec) Time (sec) (c) Step response C (d) Step response D Step Response Step Response Amplitude 1 Amplitude Time (sec) Time (sec) (e) Step response E (f) Step response F Figur 8: Step responses in Problem 4 (b). The scale is the same in all figures. 8

11 Bode Diagram Bode Diagram (a) G c1 (b) G c Bode Diagram Bode Diagram (c) G c3 (d) G c4 Bode Diagram Bode Diagram (e) G c5 (f) G c6 Figur 9: Bode plots for the closed-loop systems G c (s) = (b). The scale is the same in all figures. Go(s) 1+G o(s) in Problem 4 9

12 Bode diagram Bode diagram (a) G o1 (b) G o Bode diagram Bode diagram (c) G o3 (d) G o4 Bode diagram Bode diagram (e) G o5 (f) G o6 Figur 1: Bode plots for the open-loop systems G o (s) in Problem 4 (b). The scale is the same in all figures. 1

13 5. The model of a wind turbine can be described by the following transfer function G(s) = Y (s) U(s) = 1 τs + 1 Kω n s + ζω n s + ω n where τ = 5, K = 7, ζ =.5, ω n =, and the input u(t) is the angle of the rotor blades, the output y(t) is the rotational speed of the wind turbine. The Bode plot of G(s) is shown in Figure 11. (a) Assume that the system is controlled according to Figure 1. Determine a regulator F (s) such that the following design specifications for the open-loop system F (s)g(s) are fulfilled: The steady state error satisfies that lim t e(t) =.15 when the reference signal is a unit step signal (i.e., its amplitude is 1). The desired gain crossover frequency ω c =.4 rad/s The desired phase margin φ m 6 (8p) (b) Assume that the system is controlled using the controller designed in a) above. Determine S(iω) for the angular frequencies ω = rad/s. (p) 11

14 Bode Diagram Magnitude (abs) Frequency (rad/s) Bode Diagram Frequency (rad/s) Figur 11: Bode plot for problem 5. 1

15 Figur 1: Feedback control of the wind turbine 13